The push for a semiconductor-based quantum information technology has renewed interest in the spin states and optical transitions of shallow donors in silicon, including the donor bound exciton transitions in the near-infrared and the Rydberg, or hydrogenic, transitions in the mid-infrared. The deepest group V donor in silicon, bismuth, has a large zero-field ground state hyperfine splitting, comparable to that of rubidium, upon which the now-ubiquitous rubidium atomic clock time standard is based. Here we show that the ground state hyperfine populations of bismuth can be read out using the mid-infrared Rydberg transitions, analogous to the optical readout of the rubidium ground state populations upon which rubidium clock technology is based. We further use these transitions to demonstrate strong population pumping by resonant excitation of the bound exciton transitions, suggesting several possible approaches to a solid-state atomic clock using bismuth in silicon, or eventually in enriched 28Si.
A silicon crystal provides an extremely clean environment within which electrons may orbit impurity atoms, analogous to isolated atoms held in vacuum traps1,2,3,4. The hyperfine splittings of alkali atoms in vacuum are widely used as microwave frequency standards5, while the push for semiconductor-based quantum information technologies has resulted in dramatic improvements of the coherence times for hyperfine states of shallow donors in silicon3,6. Magnetic resonance detection in alkali atoms is enabled by selective pumping of the D-lines7, but the hyperfine splittings of the equivalent transitions to the odd-parity excited states of shallow donors in semiconductors have so far eluded detection. Here we demonstrate well-resolved hyperfine splittings of these transitions in natural silicon doped with bismuth and use these transitions to observe strong, controllable polarization of the hyperfine states by pumping the donor bound exciton transitions. Our results open the possibility of sub-ns pumping and bring solid-state frequency standards and masers based on these donor transitions within reach.
The last fifty years have been a period of outstanding developments in the field of atomic frequency standards, reaching frequency stabilities in the 10−16 range in hydrogen masers and accuracies better than 10−15 in cesium fountains5, ytterbium lasers8 and even 10−18 in a strontium lattice9. In many implementations, hydrogen or alkali-metal atoms with a single valence electron are used and the resonance transition between the two hyperfine levels of the 2S1/2 ground-state serves as the frequency reference. The splitting, which is in the microwave range (1–10 GHz), is produced by the interaction between the electron spin and the nuclear spin. The splitting is much smaller than kBT (except at mK temperature) and a common characteristic of the preparation to enhance the detected magnetic resonance signal in those implementations is the use of state selection by means of optical pumping10. The simplest “intensity pumping” scheme involving a third, higher energy state, uses repeated cycles of selective excitation followed by random relaxation to drive population away from the component being pumped. In more sophisticated variants the pumping is a coherent sum of two fields in a classic “Lambda scheme”. The oft used pumping transitions are the D-lines, i.e. 2S1/2–2P1/2 or 2S1/2–2P3/2 (D1 or D2 respectively). These excited states are separated by the fine structure introduced by the coupling of the electron spin with its orbital magnetic moment. Each of the excited levels is further split by the hyperfine interaction, though by a smaller amount than the ground state. Diode lasers with appropriate wavelength and line-width are readily available7. In this work we investigate the analogous optical pumping in silicon.
Silicon has found uses in metrology for its mechanical properties as the cavity for mHz line-width lasers11 and in parallel, electromagnetic control over impurities in silicon is advancing rapidly. Group V donors in silicon have an absorption spectrum that is closely analogous with hydrogenic free atoms because they also have a single electron that is loosely bound to a singly positively charged core. The main difference in silicon is that the Coulomb force is screened by the dielectric constant of the host and the effective mass of an electron moving in the host is less than the free electron rest mass, reducing the energy scale for the Rydberg transitions to the mid-infrared and increasing the orbital radius to a few nm4. In addition, the anisotropic, multivalley Si conduction band lifts some degeneracies so that e.g. the 2p0 state is separated from the 2p±.
The 1s(A1) ground state of group V donors in Si is split by the hyperfine interaction between the electron spin (S) and the nuclear spin (I), with Si:Bi (I = 9/2) having the largest zero-field splitting at 30.51 μeV (7.4 GHz)12,13,14. Si:P (I = 1/2) has so far exhibited the longest dephasing times of the hyperfine split components, of order 100 sec when the host is isotopically purified 28Si (I = 0) so as to remove the 29Si (I = 1/2) present in natural Si3,6. In those studies the hyperfine components were polarized by pumping with photons much above the donor ionization threshold, but resonant with the donor-bound-exciton (D0X) energy. If the neutral donor (D0) is thought of as an analog of hydrogen, then an exciton is the semiconductor equivalent of positronium and D0X is analogous with positronium hydride. It is interesting to note that bound excitons were predicted on the basis of this analogy15 well before they were first observed16. While the transitions of positronium hydride are not a useful tool for studying the energy levels of hydrogen, in semiconductors the D0X transitions lie just below the band gap energy, in the near-infrared for Si and are in many ways more accessible than the Rydberg transitions, which lie in the mid- to far-infrared.
The transitions and energy levels relevant to this study of Si:Bi are shown in Fig. 1. The hyperfine coupling splits the D0 ground state into two levels having total spin F = I + S = 5 and F = I–S = 4, separated by 30.51 μeV. The D0X ground state has no hyperfine coupling, since the exclusion principle requires the two electrons to form a spin singlet, which cannot couple to the nuclear spin. The D0X transitions can be observed at identical energies in emission via photoluminescence (PL), or in absorption, as shown by the spectra on the left hand side of Fig. 1. Due to the indirect band gap of Si, D0X absorption is very weak17 and the radiative lifetime is very long, with the decay of D0X almost completely dominated by nonradiative Auger decay, or autoionization18. We therefore measured the D0X absorption spectrum indirectly, detecting the carriers generated by the Auger decay of the created D0X, using photoconductivity (PC).
Since the hyperfine splitting occurs in the final state for D0X emission, the relative intensity of the two components in PL simply reflects the difference in degeneracies of the two hyperfine states (11:9). The relative intensities of the D0X hyperfine components in absorption reflects the relative populations in the two D0 hyperfine initial states and if the two populations were in thermal equilibrium at our temperature of 1.5 K, we would have expected the relative intensities to be near unity in the PC spectrum, since the Boltzmann factor nearly cancels the differences in degeneracy. The essentially identical relative intensities seen in the PL and PC spectra indicates that the D0 hyperfine states are not in thermal equilibrium under the conditions used to measure the PC spectrum, but are characterized by a higher effective temperature. This likely results from nonresonant photoionization of the D0 hyperfine states and recapture of the free electrons, which acts to randomize, or saturate, the two populations.
The Rydberg transitions to the odd-parity orbital excited states (D0*) form a series converging to the Bi ionization energy, 70.98 meV19. While the hyperfine splitting of the Bi ground state in the D0X transitions of Fig. 1 has already been observed14, all previous studies of the Rydberg transitions lacked sufficient spectral resolution to have resolved this splitting19,20. In the Rydberg absorption spectrum shown in Fig. 2, structure due to the Bi hyperfine ground state splitting is seen in all of the transitions. The observed splitting is ΔE = 30.5 μeV, in agreement with previous measurements using spin resonance21 and D0X spectroscopy14. Hyperfine splittings of the D0* excited states are negligible due to their p-like orbital structure and much larger spatial extent. The relative intensities of the hyperfine components are very close to the ratios seen in the D0X PL and PC spectra in Fig. 1. The improving spectral resolution with decreasing excited final state binding energy seen in Fig. 2 is similar to effects seen22 for phosphorus and boron in 28Si and must result from an increasing lifetime and therefore decreasing lifetime broadening, for the higher Rydberg states. The 2p0 state for Si:Bi (not shown) is extremely broad due to near-resonance with optical phonons19,20, which also produces a much smaller but still significant broadening of the 2p± line by reducing the excited state lifetime. The observed series ends near principal quantum number n = 7 due to concentration broadening, i.e. when then wavefunctions are so large that those of neighbouring donors overlap. The line width of the sharpest transitions is only 9 μeV, almost identical to the D0X linewidths, indicating that in both cases the linewidths result predominantly from an inhomogeneous broadening of the shared D0 ground state energy. This results from the random isotope distribution present in natural Si, which affects primarily the D0 ground state, since both the D0X and the excited D0* wavefunctions have much larger spatial extent and see an isotopic composition nearer to average than does the compact ground state23,24.
Finally, we investigate the possibility of hyperpolarization by resonant optical pumping. The same tunable single-frequency laser that was used for D0X PC spectroscopy was tuned to either the F = 4 or F = 5 component of the D0X transition and the resulting change in ground state populations was monitored using absorption spectroscopy of the D0* Rydberg states. D0X are formed only from D0 in the hyperfine state to which the laser is tuned and due to the Auger recombination and random recapture of free electrons onto the ionized donors, population is pumped away from this state and into the other hyperfine state. Under this resonant pumping, all the hyperfine doublets seen in Fig. 2 changed in the same way, with one example detailed in Fig. 3. When the laser was tuned to the F = 4 hyperfine component, the Rydberg transitions having the same ground state were reduced and the other components were correspondingly increased. The situation reversed when pumping the F = 5 component. The relative population in the two D0 hyperfine states was changed by ± 40% as a result of this optical pumping.
We turn now to possible implications and future directions. We have used the D0X transitions for polarization and the D0* transitions for readout, simply because we do not at present have a single frequency source at the D0* energies. However, the D0* energies are very close to standard III-V Quantum Cascade Laser wavelengths (commercially available down to ~80 meV) and well within the range of other solid state laser sources based on difference frequency generation25 or air-plasma generation26 (techniques with wavelength ranges that overlap). Unlike the D0X transitions, the dipole moment matrix element for the D0* transitions is very large (〈x〉~0.28 nm for the 2p0 transition2) and larger than those of free alkali atoms. We would expect a similar hyperpolarization process when resonantly pumping a D0* hyperfine component, even though this would not lead to autoionization as for pumping D0X, since memory of the original hyperfine state would be lost during the relaxation cascade from the D0* state back to the D0 ground states. The short lifetime of the D0* states, ~200 ps or less1,27, compares favorably with the ns to 100’s of ns lifetimes of the D0X18,28. While we have used both transitions in this study, an atomic clock based on the Bi ground state splitting could use either the D0X or the Rydberg transitions for both state preparation and readout.
While our observed optical polarization is already quite large it should not represent the ultimately achievable limit. As discussed previously, we believe that the linewidths of both the D0X and D0* optical transitions is limited by inhomogeneous isotope broadening inherent to natural Si, which can be essentially eliminated using highly enriched 28Si as has been shown before for the optical transitions associated with the shallower donor phosphorus and the shallow acceptor boron24. The achievable zero field Bi polarization in natural Si is limited by the mismatch between the very narrow (sub MHz) laser linewidth and the ~2 GHz inhomogeneously broadened D0X absorption linewidth. Note that the inhomogeneous broadening of these optical transitions results primarily from the mass differences between the Si isotopes and not from the nonzero nuclear spin of 29Si24. Besides making both the D0X and D0* optical transitions much sharper, using enriched 28Si will have the further advantage of greatly decreasing the linewidth of the microwave transitions between the D0 1s(A1) hyperfine components by eliminating the random magnetic fields due to 29Si nuclei which limit Bi magnetic resonance linewidths to ~500 kHz in natural Si13,29.
The density of donors in our sample is far larger than in typical atomic clock/maser applications. The latter is necessarily low due to the requirement for low collision frequency, whereas the impurities here are fixed in space without the need for high vacuum systems or trapping. The disadvantage of the silicon system is that low temperatures must be used to keep the donors neutral and reduce phonon interactions. Given the rapid advances being made in QCLs and closed-cycle coolers, it seems reasonable to speculate that large hyperpolarized populations could be created in a small system and that solid state masers and frequency standards might be within reach. It is also important to note that the Rydberg transitions used here provide electric-dipole readout of the coupled electron and nuclear spins, which is important for quantum information applications. This can be many orders of magnitude faster than magnetic dipole readout with microwaves. At the same time, the results point the way to use of Rydberg pumping to produce the polarization, which should be several orders of magnitude faster than the presently employed D0X pumping, due to the stronger absorption and faster relaxation.
Materials and Methods
All experiments were carried out with a sample from the same crystal used in the previous PL study14. It is dislocation-free float-zone grown natural Si doped with Bi with concentration 2 × 1014 cm−3 in a wafer 1 mm thick and 10 mm in diameter, cut perpendicular to the  crystal growth axis and etched in HF/HNO3 to remove surface damage. For the noncontact PC measurements3,6 the sample was mounted in a strain-free manner between copper foil electrodes in superfluid He at ~1.5 K and excited with a tunable single-frequency Yb-doped fiber laser. The PL spectrum was obtained with 532 nm above-gap excitation of the sample mounted strain-free in superfluid He at ~1.5 K, using a Fourier transform infrared (FTIR) spectrometer and a liquid nitrogen-cooled Ge p-i-n diode detector and a spectral resolution of 2.5 μeV (0.02 cm-1).
The Rydberg absorption spectrum was obtained using FTIR transmission spectroscopy with resolution 2.5 μeV (0.02 cm-1), a black body source and a liquid helium cooled Si:B photoconductive detector22. The sample was mounted strain-free in superfluid He at 1.5 K and illuminated on one edge with 1047 nm light from a diode pumped Nd:YLF laser, which is just above the band edge of silicon and produces a small population of free electrons and holes that neutralise ionized donor and acceptor impurities, reducing the consequent random local electric fields and thus improving the observed line-widths, especially for the higher excited states. For the optical pumping measurements, the sample was also illuminated with the output of the Yb-doped fiber laser tuned to either the F = 4 or F = 5 transition of the Bi D0X. The data used in this work are available to download at 10.5281/zenodo.16940
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This work was conducted at SFU and supported by the Natural Sciences and Engineering Research Council of Canada. BNM and KLL are grateful for support from the UK EPSRC (COMPASSS, grant reference EP/H026622) and BNM is also grateful for a Royal Society Wolfson Research Merit Award.
The authors declare no competing financial interests.
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Saeedi, K., Szech, M., Dluhy, P. et al. Optical pumping and readout of bismuth hyperfine states in silicon for atomic clock applications. Sci Rep 5, 10493 (2015). https://doi.org/10.1038/srep10493
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